Proof for n=4 using Gaussian Integers

In today's blog, I will show how it is possible to use Gaussian Integers to establish a proof for Fermat's Last Theorem n = 4. Previously, I showed two proofs for this result. I showed a standard proof based on Pythagorean Triples (see here). Also, I showed how it could be derived from Fermat's proof that a right triangle cannot have a square area (see here).

The proof for the case n = 4 using Gaussian Integer is both very strange and very interesting. The proof that I present is taken from Paul Ribenboem's book Fermat's Last Theorem for Amateurs.

Please note that I use Greek letters to represent Gaussian Integers and Latin letters to represent rational integers. This is really to emphasize that reasoning with Gaussian Integers requires different assumptions. For example, the Well Ordering Principle no longer applies and 2 is no longer a prime.

Theorem: x⁴ + y⁴ = z² has no solution in Gaussian Integers where xyz ≠ 0.

(1) First, I will show that if there is a solution in rational integers, then the following is true:

There exists α, β, γ, ε, λ such that:
ε * λ⁴ⁿ * α⁴ + β⁴ = γ² where n ≥ 2. [See here for proof.]

(2) Second, I will show that if (1) is true, then there is another set of values: ε₁, α₁, β₁, γ₁ such that:

ε₁ * λ^4(n-1) * α₁⁴ + β₁⁴ = γ₁². [See here for proof.]

(3) But then, repeating step (2) again and again, we eventually get to values: α_i, β_i, ε_i, γ_i where

ε_i * λ⁴*α_i⁴ + β_i⁴ = γ_i²

(4) Which contradicts step (1) where n ≥ 2. So we reject that there is a solution to FLT for n=4.

QED